Analytic methods applied to a sequence of binomial coefficients

In previous work we have shown that the binomial coefficients C_n··kr, r are strongly logarithmically concave for 0?r?n/(k+1)] and hence have at most a double maximum. Let r_{n, k} be the least integer at which this maximum occurs. Properties of {r_{n, k}}_{n, k} are best derived by introducing the polynomial family Q_k(x,y) defined by Q_k(x,y) = ∏^k+1_j=1 1?(k+1)x+jy]?x∏^k_j=11?kx+jy]. It is shown that for each k there is a unique function η_k(y) defined on 0, 1/(k+1)] which is analytic in a neighbourhood of zero and which satisfies Q_k(η_k(y), y)=0. Setting η_k(0)=δ_k, η¹_k(0)=α_k we prove that r_rn,k = nδ_k] onδ_k]+1 and further, that the number of times that r_m,k = mδ_k]+1 for k+1? m ? n is asymptotically nα_k. Several other properties of α_k are derived, including 0<α_k <1/2.